1. approximate the area under the curve f(x)=x² + 1 and above the x - axis on the interval 2,8: a) using…

Explanation:

Step1: Calculate width of sub - intervals for 3 rectangles

The interval is $[a,b]=[2,8]$, and $n = 3$. The width of each sub - interval $\Delta x=\frac{b - a}{n}=\frac{8 - 2}{3}=2$. The sub - intervals are $[2,4]$, $[4,6]$, $[6,8]$. The mid - points are $x_1 = 3$, $x_2=5$, $x_3 = 7$.

Step2: Calculate the function values at mid - points

$f(x)=x^{2}+1$. $f(x_1)=f(3)=3^{2}+1=9 + 1=10$. $f(x_2)=f(5)=5^{2}+1=25 + 1=26$. $f(x_3)=f(7)=7^{2}+1=49+1=50$.

Step3: Calculate the area using mid - point rule for 3 rectangles

$A_3=\sum_{i = 1}^{3}f(x_i)\Delta x=\Delta x(f(x_1)+f(x_2)+f(x_3))=2(10 + 26+50)=2\times86 = 172$.

Step4: Calculate width of sub - intervals for 6 rectangles

The interval is $[a,b]=[2,8]$, and $n = 6$. The width of each sub - interval $\Delta x=\frac{b - a}{n}=\frac{8 - 2}{6}=1$. The sub - intervals are $[2,3]$, $[3,4]$, $[4,5]$, $[5,6]$, $[6,7]$, $[7,8]$. The mid - points are $x_1=2.5$, $x_2 = 3.5$, $x_3=4.5$, $x_4=5.5$, $x_5=6.5$, $x_6=7.5$.

Step5: Calculate the function values at mid - points

$f(x_1)=f(2.5)=(2.5)^{2}+1=6.25 + 1=7.25$. $f(x_2)=f(3.5)=(3.5)^{2}+1=12.25 + 1=13.25$. $f(x_3)=f(4.5)=(4.5)^{2}+1=20.25 + 1=21.25$. $f(x_4)=f(5.5)=(5.5)^{2}+1=30.25 + 1=31.25$. $f(x_5)=f(6.5)=(6.5)^{2}+1=42.25 + 1=43.25$. $f(x_6)=f(7.5)=(7.5)^{2}+1=56.25 + 1=57.25$.

Step6: Calculate the area using mid - point rule for 6 rectangles

$A_6=\sum_{i = 1}^{6}f(x_i)\Delta x=\Delta x(f(x_1)+f(x_2)+f(x_3)+f(x_4)+f(x_5)+f(x_6))$ $=1\times(7.25+13.25+21.25+31.25+43.25+57.25)=173.5$.

Answer:

a) 172 b) 173.5